Multiplicative independence of modular functions

We provide a new, elementary proof of the multiplicative independence of pairwise distinct $\mathrm{GL}_2^+(\mathbb{Q})$-translates of the modular $j$-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For modular functions $f \in \overline{\mathbb{Q}}(j)$ belonging to this class, we deduce, for each $n \geq 1$, the finiteness of $n$-tuples of distinct $f$-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber--Pink conjecture for subvarieties of the mixed Shimura variety $Y(1)^n \times \mathbb{G}_{\mathrm{m}}^n$ and prove some special cases of this conjecture.